Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks

• 成果类型: Article
• 署名作者: Kabluchko, Zakhar; Marynych, Alexander
• 署名单位: University of Munster; Ministry of Education & Science of Ukraine; Taras Shevchenko National University of Kyiv
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01146-9
• 发表日期: 2022
• 页码: 969-1028
• 关键词: mod-gaussian convergence; LIMIT-THEOREMS; phase-transitions; asymptotics; simplices; polytopes; neighborliness; faces

摘要

Let xi 1, xi 2, ... be a sequence of independent copies of a random vector in R-d having an absolutely continuous distribution. Consider a random walk S-i := xi(1) + ... + ... + xi(i), and let C-n,C-d := conv(0, S-1, S-2, ..., S-n) be the convex hull of the first n+ 1 points it has visited. The polytope C-n,C-d is called k-neighborly if for any indices 0 <= i(1) < ... < i(k) <= n the convex hull of the k points S-i1 , ..., S-ik is a (k - 1)-dimensional face of C-n,C-d. We study the probability that C-n,C-d is k-neighborly in various high-dimensional asymptotic regimes, i.e. when n, d, and possibly also k diverge to infinity. There is an explicit formula for the expected number of (k - 1)-dimensional faces of C-n,C-d which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties. In particular, we provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. Limit theorems which we prove for the Lah distribution imply neighborliness properties of C-n,C-d. This yields a new class of random polytopes exhibiting phase transitions parallel to those discovered by Vershik and Sporyshev, Donoho and Tanner for random projections of regular simplices and crosspolytopes.